# Worksheet: Linear Transformation Composition

In this worksheet, we will practice finding the matrix of two or more consecutive linear transformations.

Q1:

Find the matrix with respect to the standard basis vectors for the linear transformation that rotates every vector in through an angle of and then reflects it across the -axis.

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Q2:

A linear transformation is formed by reflecting every vector in in the -axis and then rotating the resulting vector through an angle of . Find the matrix of this linear transformation.

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Q3:

A linear transformation is formed by rotating every vector in through an angle of , reflecting the resulting vector in the -axis, and finally reflecting this vector in the -axis. Find the matrix of this linear transformation.

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Q4:

A linear transformation is formed by rotating every vector in through an angle of and then reflecting the resulting vector in the -axis. Find the matrix of this linear transformation.

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Q5:

A linear transformation is formed by reflecting every vector in across the -axis and then rotating the resulting vector through an angle of . Find the matrix of this linear transformation.

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Q6:

A linear transformation is formed by rotating every vector in through an angle of and then reflecting the resulting vector in the -axis. Find the matrix of this linear transformation.

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Q7:

A linear transformation is formed by reflecting every vector in in the -axis and then rotating the resulting vector through an angle of . Find the matrix of this linear transformation.

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Q8:

A linear transformation is formed by reflecting every vector in in the -axis and then rotating the resulting vector through an angle of . Find the matrix of this linear transformation.

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Q9:

Let the matrix represent rotation in the plane through an angle of and let the matrix represent reflection in the -axis.

What is the matrix ?

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What is the matrix ?

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What is the matrix ?

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Q10:

Suppose that the matrix represents rotation about the origin through an angle of (measuring between and ) and the matrix represents reflection in the -axis.

Find .

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Note that is a reflection in a line through the origin. Let this line of reflection have equation . By considering the image of the vector , determine the measure of the angle between the -axis and the line of reflection.

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What, therefore, is the slope of the line of reflection?

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If lies in the direction of the line of reflection, what is ?

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By solving the equation obtained in the previous part, find another expression for the slope of the line of reflection.

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Q11:

A linear transformation is formed by rotating every vector in through an angle of counterclockwise (when viewed from the positive -axis) about the -axis and then reflecting the resulting vector in the -plane. Find the matrix of this linear transformation.

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Q12:

A linear transformation is formed by rotating every vector in through an angle of and then reflecting the resulting vector in the -axis. Find the matrix of this linear transformation.

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Q13:

A vector in is rotated counterclockwise about the origin through an angle of , and the result is reflected in the -axis. Find, with respect to the standard basis, the matrix of this combined transformation.

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Q14:

Suppose and are matrices, with representing a counterclockwise rotation of about the origin and representing a reflection in the -axis. What does the matrix represent?

• Aa reflection in the line through the origin at a inclination
• Ba reflection in the line through the origin at a inclination
• Ca reflection in the line through the origin at a inclination
• Da reflection in the line through the origin at a inclination
• Ea reflection in the line through the origin at a inclination

Q15:

Suppose and are matrices, with representing a counterclockwise rotation of about the origin and representing a reflection in the -axis. What does the matrix represent?

• Aa reflection in the line through the origin at a inclination
• Ba reflection in the line through the origin at a inclination
• Ca reflection in the line through the origin at a inclination
• Da reflection in the line through the origin at a inclination
• Ea reflection in the line through the origin at a inclination

Q16:

Describe the geometric effect of the transformation produced by the matrix .

• Aa dilation with center the origin and scale factor 3 followed by a reflection in the line
• Ba dilation with center the origin and scale factor 3 followed by a rotation about the origin
• Ca dilation with center the origin and scale factor 3 followed by a reflection in the line
• Da dilation with center the origin and scale factor 3 followed by a rotation about the origin
• Ea dilation with center the origin and scale factor 3 followed by a rotation about the origin

Q17:

Which of the following compositions of transformations is represented by the matrix ?

• Aa dilation with center the origin and scale factor 2 followed by a reflection in the line
• Ba rotation by about the origin followed by a reflection in the line
• Ca dilation with center the origin and scale factor 2 followed by a reflection in the line
• Da dilation with center the origin and scale factor followed by a reflection in the line
• Ea dilation with center the origin and scale factor followed by a reflection in the line

Q18:

A dilation with center the origin is composed with a rotation about the origin to form a new linear transformation. The transformation formed sends the vector to .

Find the matrix representation of the transformation formed.

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Find the scale factor of the original dilation.

• Ascale factor = 169
• Bscale factor = 13
• Cscale factor = 154
• Dscale factor = 13
• Escale factor =

Q19:

The unit square, with vertices , and , is transformed by a rotation and then a dilation. Its image under this combined transformation is , as shown in the diagram. What are the coordinates of ?

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What is the matrix of the combined transformation?

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